Orbits of $\QQ^*(\sqrt{n})$ under the action of Modular Group $PSL(2,\mathbb{Z})$

Abstract

Coset diagrams have been used to study quotients, orbits, subgroups and structure of the finitely generated groups. In this paper we use coset diagrams and modular arithmetic to determine the $G$-orbits of $\QQ^*(\sqrt{p^k})$, $\QQ^*(\sqrt{2p^k})$, $\QQ^*(\sqrt{2^2p^k})$, and in general $\QQ^*(\sqrt{2^lp^k})$, for each $l\ge 3$ and $k=2h+1\ge 3$, for each odd prime $p$.

arXiv commentThis paper includes 9 pages and 12 references